While you have a point, the golden rule is to shut down everything related to email and IM during meetings.

But you can't control what other people do. You can control what messages you send.

But you can expect people to do their jobs. If a notification embarrassed the company in front of a customer then talk to that person's manager about training them to not be a numpty.

Stop having such low expectations of grown ass adults.

You clearly haven't worked at a job with incompetent adults before. I mostly haven't either thankfully, but my wife just put in her two weeks resignation after two months at her new job because she works proposal management and she couldn't get anyone to do anything properly (proposal management requires getting information from other people in order to put together a proposal with all the information requested), while they just kept piling enough work to keep at least 3 employees busy on her.

Forms filled out wrong after explicit instructions, vital information needed for a proposal due at 10 in the morning not received until 5pm the day before (and even then it's missing half of what they were told was needed), and then after asking for it again, finding out half an hour before it's due "oh I'm out running an errand, I'm not in the office, I can't get that to you", people taking pictures of handwritten notes in sloppy handwriting and sending them to other coworkers instead of typing them up themselves, people refusing to click a link on their iPads to access a document and demanding email attachments instead, people refusing to store important documents in CRM tools and instead saying "well it's in someone's email somewhere".

She's worked with several of these types before, but not this many people at one company, and not with such an intense workload (I think she was expected to submit a proposal every other day this month while wrangling these people, which is very short. At past jobs she usually only had to juggle 1 or 2 proposals in any given week).

She had no choice but to work holidays and nights and weekends and it still looked like she wasn't doing a good job because they weren't doing their jobs (her boss knew she was though, since she had to do the job before she was hired and knew what it was like, and begged her to stay).

I guarantee none of these people would have bothered turning off a notification, and then something confidential (legally not supposed to be seen by certain employees) could have been revealed.

Not that I ever usually bother to think about that myself, personally. It's rare that I start a conversation with someone with 'Hey blah! Here's some information that I can get in trouble if other people besides you see!'

It's called snooze your notifications. Easy enough to do, at least on macOS, and surely all OSs with a unified notification manager.

You can control what messages you receive by closing your client

There may be a middle ground. In this example, you could say, "Are you available to join a call at 3:30?"; that gives a little information about the topic and the time sensitivity, but it may still require a follow-up.